Phase, Geodesics, and Space-Time Curvature

Please read and critique this argument for me please, any help is appreciated.

Imagine a geodesic, and a matter wave that traverses this geodesic. The action of this matter wave determines the motion of the matter wave along this geodesic over a given space-time interval, and is specified by a metric and infinitesimal space-time lengths. The phase of the matter wave evolves along this geodesic, space-time interval.

Over a given space-time interval, there exist multiple geodesics along which a matter wave can traverse. This is done by holding the end-points of the space-time interval fixed, while varying the metric signature and space-time lengths. For each one of these geodesics, the matter wave's phase has the same initial value, but over the different geodesics evolves to different values with respect to each geodesic.

Therefore, a relationship between the phase of a matter wave, and its evolution along a geodesic should be derivable. A sum over all the phases (and hence the geodesics) would leave one with a prominent phase (or phase distribution) which could in turn be matched with a corresponding geodesic (over a fixed and given space-time interval, of which there should be a one-to-one correspondence between the phase and geodesic path). This geodesic can then be used to find the curvature of space-time (because of its intrinsic connection to curvature through the metric signature).